Getting thyroid hormone levels and nutrient levels optimised will help the problems you list. But it might be worth experimenting with going gluten-free for 8 - 12 weeks or so as well.
Lots of people on here get benefits from going gluten-free. But in most cases it is because it reduces thyroid antibodies. I have never had thyroid antibodies, and I've been tested a few times. But eventually, reluctantly, I decided to try going gluten-free as an experiment. And the one really, really obvious thing that improved in less than a week was my irritation and temper.
So, if the irritability is becoming a problem, consider going gluten-free. It might help, you never know. And if it doesn't you can always start eating it again.
I have been reading about Armour.
Just for info...
Armour Thyroid is a brand name of NDT (Natural Dessicated Thyroid). It is one of several brands of NDT available.
NDT has an active ingredient which usually comes from pig thyroid, although I have read about some which contain cow thyroid but I don't know any brand names for cow thyroid pills.
Armour Thyroid is the most expensive brand of NDT in the world. Price is not a guide to quality. Many people do just fine on the cheapest.
Other brands of NDT available are listed here :
Click where it says "Non-UK - Desiccated Thyroid (aka NDT)"
There are some brands of NDT not listed on that link - the ones from Thailand. They are Thiroyd, Thyroid-S, and TR Man. Many people do just fine on the Thai brands, which are the cheapest available.
Having said all the above... If you decide to try NDT and then mention it to a doctor, always, always refer to it as Armour, no matter which brand you take. If the doctor has actually heard of NDT (which is not guaranteed at all) for some reason the only name they recognise is Armour and anything else will be dismissed as dodgy. Don't ask me why.